AI summaryⓘ
The authors explore how a learning system can look the same right now based on its current behavior, but still react differently when it learns in the future. They introduce a concept called fiber fingerprints to describe how future learning responses vary even when current behaviors appear identical. Their work uses mathematical tools to break down these responses into parts that are visible, reused, or new, without relying on common assumptions like smoothness or reversibility. They also test their ideas on recent AI models, finding complex patterns in how these models update during training. Overall, the authors show that just looking at a system's present behavior isn't enough to predict how it will learn going forward.
fiber fingerprintspresent-behavior equivalencepredictive quotient functorNerode equivalenceHilbert space realizationgraph-Hodge decompositionAdam optimizerTransformer modelsmoment sectionnon-closure
Abstract
A learning system can occupy execution states that are indistinguishable under every declared present-behavior readout yet respond differently to future training. We formalize this through fiber fingerprints: controlled future-learning response laws restricted to present-behavior equivalence classes. Prefix-compatible finite probes induce a predictive quotient functor, a Nerode-type minimal recursively sufficient representation, and a canonical set-level predictive fiber without assuming smoothness, reversibility, finite rank, or a manifold. Under an explicit finite-dimensional Hilbert realization, response decomposes into visible, visible-mode-reuse, and irreducible-new sectors; a history-reachability bridge retains only distinctions generated by natural training histories. Conditional mechanism results then identify a graph-Hodge chronology decomposition, a regular switching class with root-mean-square scale $\sqrt{p}η^{3/2}$ and finite-scale corrections, and an exact Adam moment section whose immediate adaptive field is constant while common future gradients can reveal hidden moment differences. Frozen Transformer--LoRA--AdamW studies with Qwen2.5-7B and Mistral-7B-v0.3 support a local action backbone, longer-horizon first-return non-closure, and fresh visible-relative completion with output-range reuse and a low-rank irreducible sector. Stronger claims remain bounded by preregistered negative or mixed results: re-anchored transport is unresolved above its measurement floor; the strict finite-grid Hodge--$3/2$ conjunction is unmet despite prospective contraction; Qwen accessibility is not established in the frozen raw moment chart; and Mistral revelation is future-context dependent rather than bank invariant. Within these support-, scale-, metric-, and context-resolved boundaries, present behavior is not a sufficient statistic for declared future learning.